Narrow Results

Let G be a finite group and p a prime. We denote by ${\mathrm{\text{Irr}}}_p(G)$ the set of irreducible complex characters of G whose degrees are linear or divisible by p, and we write ${\mathrm{\text{R}}}_p(G)$ to denote the ratio of the sum of squares of irreducible character degrees in ${\mathrm{\text{Irr}}}_p(G)$ to the sum of irreducible character degrees in ${\mathrm{\text{Irr}}}_p(G)$. The Itô–Michler Theorem on character degrees states that ${\mathrm{\text{R}}}_p(G)=1$ if and only if G has a normal abelian Sylow p-subgroup. We generalize this theorem as follows: if ${\mathrm{\text{R}}}_p(G)<\frac{p+1}{2}$, then G has a normal Sylow p-subgroup.

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

Let $P$ be a Sylow $p$-subgroup and $b(G)$ be the largest irreducible character degree of a finite nonabelian group $G$. Then $|P/O_p(G)|\le {(b{(G)}^p/p)}^{\frac{1}{p-1}}$.

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra357 (2012) 61–68] the following question arose:

Question.Which groups can be uniquely determined by the structure of their complex group algebras?

It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of $ℂL$, the complex group algebra of $L$. One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if $p>3$ is an odd prime number, then $\mathrm{\text{PSL}}(2,p)\times \mathrm{\text{PSL}}(2,p)$ is uniquely determined by the structure of its complex group algebra.

Let $G$ be a finite nonabelian group, let $f(G)$ be the minimal degree of a nonlinear irreducible character of $G$ and suppose that $\left|G^{n+1}\right|\le f(G)$ for some positive integer $n$. Then $G$ is nilpotent.

Let $S$ be a Suzuki group ${}^2{B}_2(q^2)$, where $q^2=2^{2f+1}$, $f\ge 1$. In this paper, we determine the degrees of the ordinary complex irreducible characters of every group $G$ such that $S\le G\le \mathrm{\text{Aut}}(S)$.

In this paper, we prove that the direct product $\mathrm{\text{Sz}}(q_1)\times \mathrm{\text{Sz}}(q_2)\times \cdots \times \mathrm{\text{Sz}}(q_k)$, where $q_i=2^{2n_i+1}\ge 8$ are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product $\mathrm{\text{Sz}}(2^{p_1})\times \mathrm{\text{Sz}}(2^{p_2})\times \cdots \times \mathrm{\text{Sz}}(2^{p_k})$, where $p_i$’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

For a finite group $G$, let $\mathrm{\Delta }(G)$ be the character-graph which is built on the set of irreducible complex character degrees of $G$. In this paper, we wish to determine the structure of finite groups $G$ such that $\mathrm{\Delta }(G)$ is $1$-connected with nonbipartite complement. Also, we classify all $1$-connected graphs with nonbipartite complement that can occur as the character-graph $\mathrm{\Delta }(G)$ of a finite group $G$.

A finite group $G$ is said to be a $T_{2,k}$-group for some integer $k\ge 1$, if there exist two distinct nontrivial degrees $a,b\in \mathrm{\text{cd}}(G)\setminus \{1\}$ such that the multiplicities of $a$ and $b$ are $2$ and $k$, respectively, and for every $d\in \mathrm{\text{cd}}(G)\setminus \{1,a,b\}$ the multiplicity of $d$ in $G$ is trivial. In this paper, we prove that if $G$ is a nonsolvable $T_{2,k}$-group, then $k\le 4$ and we determine all nonsolvable $T_{2,k}$-groups.

Let $G$ be a finite group and the codegree of an irreducible character $\chi$ is the number $|G:\mathrm{\text{ker}}(\chi )|/\chi (1)$. In this paper, we consider nonsolvable groups with few character codegrees and prove that if a nonsolvable group $G$ has exactly four character codegrees, then $G\cong {\mathrm{\text{L}}}_2(2^f)$, where $f\ge 2$.

In this paper, we classify and determine the structure of finite groups whose character degree graphs are empty graphs.

Let $G$ be a finite group and ${\mathrm{\text{cd}}}^{\ast }(G)$ be the set of nonlinear irreducible character degrees of $G$. Suppose that $\rho (G)$ is the set of primes dividing some elements of ${\mathrm{\text{cd}}}^{\ast }(G)$. The bipartite divisor graph for ${\mathrm{\text{cd}}}^{\ast }(G)$, $B(G)$, is a graph whose vertices are the disjoint union of $\rho (G)$ and ${\mathrm{\text{cd}}}^{\ast }(G)$, and a vertex $p\in \rho (G)$ is connected to a vertex $a\in {\mathrm{\text{cd}}}^{\ast }(G)$ if and only if $p|a$. In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.